Proof of Order Facts for Natural Numbers: Not Less Than Gives Greater or Equal, and Below n+1 Gives at Most n

For (i): by Arithmetic and Order of the Natural Numbers (vii), exactly one of $m<n$, $m=n$ and $n<m$ holds. Since $m<n$ fails, $n<m$ or $n=m$, and so $n\le m$ by Arithmetic and Order of the Natural Numbers (vi), which gives $a\le b$ if and only if $a<b$ or $a=b$.

For (ii): if $m<n+1$, then $m<n$ or $m=n$ by Induction for Predicative Formulas, and the Order of ℕ below 1 and below n+1 (iii), and hence $m\le n$ by Arithmetic and Order of the Natural Numbers (vi).